186 research outputs found
Stability under Galerkin truncation of A-stable Runge--Kutta discretizations in time
We consider semilinear evolution equations for which the linear part is
normal and generates a strongly continuous semigroup and the nonlinear part is
sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow
by an implicit, A-stable Runge--Kutta discretization in time and a spectral
Galerkin truncation in space. We show regularity of the Galerkin-truncated
semiflow and its time-discretization on open sets of initial values with bounds
that are uniform in the spatial resolution and the initial value. We also prove
convergence of the space-time discretization without any condition that couples
the time step to the spatial resolution. Then we estimate the Galerkin
truncation error for the semiflow of the evolution equation, its Runge--Kutta
discretization, and their respective derivatives, showing how the order of the
Galerkin truncation error depends on the smoothness of the initial data. Our
results apply, in particular, to the semilinear wave equation and to the
nonlinear Schr\"odinger equation
Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations
We prove that a class of A-stable symplectic Runge--Kutta time
semidiscretizations (including the Gauss--Legendre methods) applied to a class
of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic
functions with analytic initial data can be embedded into a modified
Hamiltonian flow up to an exponentially small error. As a consequence, such
time-semidiscretizations conserve the modified Hamiltonian up to an
exponentially small error. The modified Hamiltonian is -close to the
original energy where is the order of the method and the time
step-size. Examples of such systems are the semilinear wave equation or the
nonlinear Schr\"odinger equation with analytic nonlinearity and periodic
boundary conditions. Standard Hamiltonian interpolation results do not apply
here because of the occurrence of unbounded operators in the construction of
the modified vector field. This loss of regularity in the construction can be
taken care of by projecting the PDE to a subspace where the operators occurring
in the evolution equation are bounded and by coupling the number of excited
modes as well as the number of terms in the expansion of the modified vector
field with the step size. This way we obtain exponential estimates of the form
with and ; for the semilinear wave
equation, , and for the nonlinear Schr\"odinger equation, . We give
an example which shows that analyticity of the initial data is necessary to
obtain exponential estimates
Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators
In recent years there has been an increasing interest in studying
time-delayed coupled networks of oscillators since these occur in many real
life applications. In many cases symmetry patterns can emerge in these
networks, as a consequence a part of the system might repeat itself, and
properties of this subsystem are representative of the dynamics on the whole
phase space. In this paper an analysis of the second order N-node time-delay
fully connected network is presented which is based on previous work by Correa
and Piqueira \cite{Correa2013} for a 2-node network. This study is carried out
using symmetry groups. We show the existence of multiple eigenvalues forced by
symmetry, as well as the existence of Hopf bifurcations. Three different models
are used to analyze the network dynamics, namely, the full-phase, the phase,
and the phase-difference model. We determine a finite set of frequencies
, that might correspond to Hopf bifurcations in each case for critical
values of the delay. The map is used to actually find Hopf bifurcations
along with numerical calculations using the Lambert W function. Numerical
simulations are used in order to confirm the analytical results. Although we
restrict attention to second order nodes, the results could be extended to
higher order networks provided the time-delay in the connections between nodes
remains equal.Comment: 41 pages, 18 figure
A-stable Runge-Kutta methods for semilinear evolution equations
We consider semilinear evolution equations for which the linear part
generates a strongly continuous semigroup and the nonlinear part is
sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the
existence of solutions which are temporally smooth in the norm of the lowest
rung of the scale for an open set of initial data on the highest rung of the
scale. Under the same assumptions, we prove that a class of implicit,
-stable Runge--Kutta semidiscretizations in time of such equations are
smooth as maps from open subsets of the highest rung into the lowest rung of
the scale. Under the additional assumption that the linear part of the
evolution equation is normal or sectorial, we prove full order convergence of
the semidiscretization in time for initial data on open sets. Our results
apply, in particular, to the semilinear wave equation and to the nonlinear
Schr\"odinger equation
Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems
In the presence of noncompact symmetry, the stability of relative equilibria
under momentum-preserving perturbations does not generally imply robust
stability under momentum-changing perturbations. For axisymmetric relative
equilibria of Hamiltonian systems with Euclidean symmetry, we investigate
different mechanisms of stability: stability by energy-momentum confinement,
KAM, and Nekhoroshev stability, and we explain the transitions between these.
We apply our results to the Kirchhoff model for the motion of an axisymmetric
underwater vehicle, and we numerically study dissipation induced instability of
KAM stable relative equilibria for this system.Comment: Minor revisions. Typographical errors correcte
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